Consistency between functional and structural networks of coupled nonlinear oscillators

In data-based reconstruction of complex networks, dynamical information can be measured and exploited to generate a functional network, but is it a true representation of the actual (structural) network? That is, when do the functional and structural networks match and is a perfect matching possible? To address these questions, we use coupled nonlinear oscillator networks and investigate the transition in the synchronization dynamics to identify the conditions under which the functional and structural networks are best matched. We find that, as the coupling strength is increased in the weak-coupling regime, the consistency between the two networks first increases and then decreases, reaching maximum in an optimal coupling regime. Moreover, by changing the network structure, we find that both the optimal regime and the maximum consistency will be affected. In particular, the consistency for heterogeneous networks is generally weaker than that for homogeneous networks. Based on the stability of the functional network, we propose further an efficient method to identify the optimal coupling regime in realistic situations where the detailed information about the network structure, such as the network size and the number of edges, is not available. Two real-world examples are given: corticocortical network of cat brain and the Nepal power grid. Our results provide new insights not only into the fundamental interplay between network structure and dynamics but also into the development of methodologies to reconstruct complex networks from data.

Figure 1

For SWNs of coupled chaotic logistic maps, [(a)–(d)] consistency between the functional (red dots) and structural (open circles) networks and (e) the probability distributions of the correlation coefficient $c_{ij}$ for different coupling strengths. The parameters are (a) $\varepsilon =1\times {10}^{-3}$, $S=0.37$; (b) $\varepsilon =2\times {10}^{-3}$, $S=0.77$; (c) $\varepsilon =1\times {10}^{-2}$, $S=1$; and (d) $\varepsilon =2\times {10}^{-2}$, $S=0.61$. The results are averaged over 20 system realizations.

Figure 2

For SW networks, the network similarity coefficient (open triangles) $S$ and the link stability coefficient (open circles) $\eta$ versus the coupling strength $\varepsilon$. Both $S$ and $\eta$ reach unity values for $\varepsilon \in [{\varepsilon }_1,{\varepsilon }_2]$, where ${\varepsilon }_1\approx 9\times {10}^{-3}$ and ${\varepsilon }_2\approx 1.5\times {10}^{-2}$. The results are averaged over 20 system realizations.

Figure 3

Network similarity coefficient (open triangles) $S$ and the link stability coefficient (open circles) versus the coupling strength $\varepsilon$ for (a) RNs and (b) SFNs. In (a), $S=\eta =1$ is reached for $\varepsilon \in [{\varepsilon }_1,{\varepsilon }_2]$, with ${\varepsilon }_1\approx 8\times {10}^{-3}$ and ${\varepsilon }_2\approx 1.2\times {10}^{-2}$. In (b), we have $S_{\max }\approx 0.985$ and ${\eta }_{\max }\approx 0.993$ for ${\varepsilon }_o=7\times {10}^{-3}$. Each data point is averaged over 20 system realizations.

Figure 4

For RNs of $N=50$ and different connection density, variations in the network similarity coefficient (open triangles) $S$ and the link stability coefficient (open circles) as a function of the coupling strength $\varepsilon$. For $\langle k\rangle =12$ (blue), we have $S_{\max }\approx 0.977$ and ${\eta }_{\max }\approx 0.937$ for ${\varepsilon }_o=7\times {10}^{-3}$. For $\langle k\rangle =20$ (red), we have $S_{\max }\approx 0.868$ and ${\eta }_{\max }\approx 0.814$ for ${\varepsilon }_o=4\times {10}^{-3}$. Each data point is averaged over 20 system realizations.

Figure 5

Network consistency in real-world networked systems. (a) For the corticocortical network of cat brain, $S$ (open triangles) and $\eta$ (open circles) versus the coupling strength. The optimal coupling strength is ${\varepsilon }_o\approx 3.5\times {10}^{-3}$. (b) For the Nepal power grid, the maximum values of $S$ and $\eta$ are achieved within the interval $\varepsilon \in [{\varepsilon }_1,{\varepsilon }_2]$, where ${\varepsilon }_1\approx 6\times {10}^{-3}$ and ${\varepsilon }_2\approx 1.1\times {10}^{-2}$. Each data point is averaged over 20 system realizations.

Figure 6

For different network models, $S$ (open triangles) and $\eta$ (open circles) versus the coupling strength: (a) RN of $N=500$ and $\langle k\rangle =6$ where we have ${\varepsilon }_o\approx 6\times {10}^{-3}$, $S_{\max }\approx 0.997$, and ${\eta }_{\max }\approx 0.961$; (b) RN of coupled chaotic Rössler oscillators, where we have ${\varepsilon }_o\approx 1.1\times {10}^{-2}$, $S_{\max }\approx 0.56$, and ${\eta }_{\max }\approx 0.94$; (c) RN of coupled noisy phase oscillators, where we have ${\varepsilon }_o\approx 7\times {10}^{-3}$, $S_{\max }\approx 0.42$, and ${\eta }_{\max }\approx 0.86$. Each data point is the result of averaging over 20 system realizations.